Lecture 5

Linear Models for Correlated Data: Inference

Inference

• Estimation Methods– Weighted Least Squares (WLS)

(Vi known)– Maximum Likelihood (Vi unknown)– Restricted Maximum Likelihood

(Vi unknown)– Robust Estimation (Vi unknown)

• Hypothesis Testing• Example: Growth of Sitka Trees

Weighted-Least Squares Estimation

Weighted-Least Squares Estimation (cont’d)

Weighted-Least Squares Estimation (cont’d)

Weighted-Least Squares Estimation (cont’d)

Weighted-Least Squares Estimation (cont’d)

Weighted-Least Squares Estimation (cont’d)

Weighted-Least Squares Estimation (cont’d)

Weighted-Least Squares Estimation (cont’d)

Estimation of Mean Model: Weighted Least Squares

Estimation of Mean Model: Weighted Least Squares (cont’d)

Estimation of Mean Model: Weighted Least Squares (cont’d)

Note that we can re-write the WRRS as:

What does this equation say?Examples…

Examples: V diagonal

Examples: V diagonal (cont’d)

Examples: V not diagonal

Examples: AR-1 (V not diagonal)

Examples: AR-1 (V not diagonal) (cont’d)

Weighted Least Squares Estimation:Summary

Pigs – “WLS” Fit

“WLS” Model results

Pigs – “WLS” Fit

Pigs – “WLS” Fit

Pigs – “WLS” Fit

Pigs – “WLS” Fit

Pigs – OLS fit. regress weight time

Source | SS df MS Number of obs = 432-------------+------------------------------ F( 1, 430) = 5757.41 Model | 111060.882 1 111060.882 Prob > F = 0.0000 Residual | 8294.72677 430 19.2900622 R-squared = 0.9305-------------+------------------------------ Adj R-squared = 0.9303 Total | 119355.609 431 276.927167 Root MSE = 4.392

------------------------------------------------------------------------------ weight | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- time | 6.209896 .0818409 75.88 0.000 6.049038 6.370754 _cons | 19.35561 .4605447 42.03 0.000 18.45041 20.26081------------------------------------------------------------------------------

OLS results

Pigs – “WLS” Fit

Pigs – “WLS” Fit

“WLS” Model results

Pigs – OLS fit. regress weight time

Source | SS df MS Number of obs = 432-------------+------------------------------ F( 1, 430) = 5757.41 Model | 111060.882 1 111060.882 Prob > F = 0.0000 Residual | 8294.72677 430 19.2900622 R-squared = 0.9305-------------+------------------------------ Adj R-squared = 0.9303 Total | 119355.609 431 276.927167 Root MSE = 4.392

------------------------------------------------------------------------------ weight | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- time | 6.209896 .0818409 75.88 0.000 6.049038 6.370754 _cons | 19.35561 .4605447 42.03 0.000 18.45041 20.26081------------------------------------------------------------------------------

OLS results

Efficiency

Efficiency (cont’d)

Example

Example (cont’d)

When can we use OLS and ignore V?

1. Uniform Correlation Model2. Balanced Data

When can we use OLS and ignore V? (cont’d)

1. (Uniform Correlation) With a common correlation between any two equally-spaced measurements on the same unit, there is no reason to weight measurements differently.

2. (Balanced Data) This would not be true if the number of measurements varied between units because, with >0, units with more measurements would then convey more information per unit than units with fewer measurements.

When can we use OLS and ignore V? (cont’d)

In many circumstances where there is a balanced design, the OLS estimator is perfectly satisfactory for point estimation.

Example: Two-treatment crossover design

Example: Two-treatment crossover design (cont’d)

Example: Two-treatment crossover design (cont’d)

Example: Two-treatment crossover design (cont’d)

(Recall slide) Inference

• Estimation Methods– Weighted Least Squares (WLS)

(Vi known)– Maximum Likelihood (Vi unknown)– Restricted Maximum Likelihood

(Vi unknown)– Robust Estimation (Vi unknown)

• Hypothesis Testing• Example: Growth of Sitka Trees

Maximum Likelihood Estimation under a Gaussian Assumption

Maximum Likelihood Estimation under a Gaussian Assumption (cont’d)

Maximum Likelihood Estimation under a Gaussian Assumption (cont’d)

Maximum Likelihood Estimation under a Gaussian Assumption (cont’d)

(Recall slide) Inference

• Estimation Methods– Weighted Least Squares (WLS)

(Vi known)– Maximum Likelihood (Vi unknown)– Restricted Maximum Likelihood

(Vi unknown)– Robust Estimation (Vi unknown)

• Hypothesis Testing• Example: Growth of Sitka Trees

Restricted Maximum Likelihood Estimation

(Recall slide) Inference

• Estimation Methods– Weighted Least Squares (WLS)

(Vi known)– Maximum Likelihood (Vi unknown)– Restricted Maximum Likelihood

(Vi unknown)– Robust Estimation (Vi unknown)

• Hypothesis Testing• Example: Growth of Sitka Trees

Generalized Least Square EstimatorRobust Estimation

(unstructured covariance matrix)

Robust Estimation of V under a saturated model

Robust Estimation of V“restricted ML” – makes estimates unbiased

Example

Robust Estimation vs.

A Parametric Approach

Maximum Likelihood Estimation of V

Example: Growth of sitka trees

Figure 1. Observed data and mean response profiles in each of the four growth chambers for the treatment and control.

Figure 2. Observed mean response in each of the four

chambers.

Season 1

Season 2

Example: Growth of sitka trees (cont’d)

Example: Growth of sitka trees (cont’d)

We first consider the 1998 data.

Example: Growth of sitka trees (cont’d)

• Unstructured covariance matrix

Example: Growth of sitka trees (cont’d)

Example: Growth of sitka trees (cont’d)

Example: Growth of sitka trees (cont’d)

Example: Growth of sitka trees (cont’d)

Sitka spruce data: Estimated covariance matrix for 1988

Sitka spruce data: Estimated covariance matrix for 1989

Summary: Unstructured Covariance Matrix

Summary: Parametric Models for Covariance

Reasons for parametric modelling:

Summary: Parametric Models for Covariance

(cont’d)Reasons for parametric modelling (cont’d):

Summary: Unstructured vs. Parametric Covariance

Overall Summary

Overall Summary